Given \(A\subset X\) , choose an increasing sequence \((K_j)\subset A\)  of compact sets such that \(c(K_j)\rightarrow c_*(A)\) ,
and let \(A^{\prime }\)  denote the union of all those \(K_j\) .
By (REF ) applied to \(A^{\prime }\)  (or alternatively by [1]}, the sets \(K_j\)  being universally measurable),
\(c_*(A^{\prime })=\lim _{j\rightarrow \infty }\,c(K_j),\) 
and hence \(c_*(A^{\prime })=c_*(A)\) . For any \(H\)  such that \(A^{\prime }\subset H\subset A\) , (REF ) now follows
from the monotonicity of the inner capacity:
\(c_*(A^{\prime })\leqslant c_*(H)\leqslant c_*(A)=c_*(A^{\prime }).\) 
Assume moreover that \(c_*(A)<\infty \) . Noting that \(\Gamma _A^+\subset \Gamma _H^+\subset \Gamma _{A^{\prime }}^+\) , we derive (REF ) from (REF ) by use of the uniqueness of the inner capacitary measure.
